To determine the interquartile range (IQR) and identify outliers, follow these steps:

Step 1: Organize the Data in Ascending Order

The given data set:

18, 24, 32, 33, 34, 35, 39, 39, 41, 43, 44, 44, 44, 46, 47, 48, 48, 48, 49, 50, 50, 50, 52, 52, 52, 52, 53, 54, 55, 55, 56, 57, 58, 58, 59, 59, 59, 60, 61, 62, 62, 62, 62, 65, 68, 72, 73, 74

There are 48 values in total.

Step 2: Find Q1 (First Quartile) and Q3 (Third Quartile)

• Q1 is the 25th percentile (the median of the first half of the data).
• Q3 is the 75th percentile (the median of the second half of the data).

Finding Q1 (Median of the first 24 values)

First half:
18, 24, 32, 33, 34, 35, 39, 39, 41, 43, 44, 44, 44, 46, 47, 48, 48, 48, 49, 50, 50, 50, 52, 52

The median of these 24 values is the average of the 12th and 13th values:
Q1 = (44 + 44) / 2 = 44

Finding Q3 (Median of the second 24 values)

Second half:
52, 52, 53, 54, 55, 55, 56, 57, 58, 58, 59, 59, 59, 60, 61, 62, 62, 62, 62, 65, 68, 72, 73, 74

The median of these 24 values is the average of the 12th and 13th values:
Q3 = (59 + 59) / 2 = 59

Step 3: Compute the Interquartile Range (IQR)

$IQR = Q3 - Q1 = 59 - 44 = 15$

Step 4: Find Outliers

An outlier is any value that falls below Q1 - 1.5(IQR) or above Q3 + 1.5(IQR).

Lower Bound:

$Q1 - 1.5(IQR) = 44 - 1.5(15) = 44 - 22.5 = 21.5$ Any value below 21.5 is an outlier.

Upper Bound:

$Q3 + 1.5(IQR) = 59 + 1.5(15) = 59 + 22.5 = 81.5$ Any value above 81.5 is an outlier.

Identifying Outliers

• The lowest value in the data set is 18, which is less than 21.5, so 18 is an outlier.
• The highest value is 74, which is less than 81.5, so it is not an outlier.

Final Answer

IQR: 15
Outlier(s): 18

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Would you like more details or clarification?

Here are 5 related questions for practice:

1. What is the difference between IQR and range?
2. How do box plots help in identifying outliers?
3. Can a data set have no outliers? If so, how?
4. How does changing a single value affect Q1, Q3, and IQR?
5. How do you interpret the presence of outliers in a dataset?

Tip: The IQR is resistant to extreme values, making it a better measure of spread than the range in skewed distributions!